Independent and Dependent Events: Understanding Probabilities

1. Independent and Dependent Events 10-7 Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 1 Holt Algebra 1

2. Warm Up Find the theoretical probability of each outcome 1. rolling a 6 on a number cube. 2. rolling an odd number on a number cube. 3. flipping two coins and both landing head up

3. Objectives Find the probability of independent events. Find the probability of dependent events.

4. Vocabulary independent events dependent events

5. Adam’s teacher gives the class two list of titles and asks each student to choose two of them to read. Adam can choose one title from each list or two titles from the same list.

6. Events are independent events if the occurrence of one event does not affect the probability of the other. Events are dependent events if the occurrence of one event does affect the probability of the other.

7. Example 1: Classifying Events as Independent or Dependent Tell whether each set of events is independent or dependent. Explain you answer. A. You select a card from a standard deck of cards and hold it. A friend selects another card from the same deck. Dependent; your friend cannot pick the card you picked and has fewer cards to choose from. B. You flip a coin and it lands heads up. You flip the same coin and it lands heads up again. Independent; the result of the first toss does not affect the sample space for the second toss.

8. Check It Out! Example 1 Tell whether each set of events is independent or dependent. Explain you answer. a. A number cube lands showing an odd number. It is rolled a second time and lands showing a 6. Independent; the result of rolling the number cube the 1sttime does not affect the result of the 2nd roll. b. One student in your class is chosen for a project. Then another student in the class is chosen. Dependent; choosing the 1st student leaves fewer students to choose from the 2nd time.

9. Suppose an experiment involves flipping two fair coins. The sample space of outcomes is shown by the tree diagram. Determine the theoretical probability of both coins landing heads up.

10. Now look back at the separate theoretical probabilities of each coin landing heads up. The theoretical probability in each case is . The product of these two probabilities is , the same probability shown by the tree diagram. To determine the probability of two independent events, multiply the probabilities of the two events.

12. Example 2A: Finding the Probability of Independent Events An experiment consists of randomly selecting a marble from a bag, replacing it, and then selecting another marble. The bag contains 3 red marbles and 12 green marbles. What is the probability of selecting a red marble and then a green marble? Because the first marble is replaced after it is selected, the sample space for each selection is the same. The events are independent.

13. The probability of selecting red is , and the probability of selecting green is . Example 2A Continued P(red, green) = P(red)  P(green)

14. The probability of landing heads up is with each event. Example 2B: Finding the Probability of Independent Events A coin is flipped 4 times. What is the probability of flipping 4 heads in a row. Because each flip of the coin has an equal probability of landing heads up, or a tails, the sample space for each flip is the same. The events are independent. P(h, h, h, h) = P(h) •P(h) •P(h) •P(h)

15. P(odd, odd) = P(odd) P(odd) • Check It Out! Example 2 An experiment consists of spinning the spinner twice. What is the probability of spinning two odd numbers? The result of one spin does not affect any following spins. The events are independent. With 6 numbers on the spinner, 3 of which are odd, the probability of landing on two odd numbers is .

16. Suppose an experiment involves drawing marbles from a bag. Determine the theoretical probability of drawing a red marble and then drawing a second red marble without replacing the first one. Probability of drawing a red marble on the first draw

17. Suppose an experiment involves drawing marbles from a bag. Determine the theoretical probability of drawing a red marble and then drawing a second red marble without replacing the first one. Probability of drawing a red marble on the second draw

18. To determine the probability of two dependent events, multiply the probability of the first event times the probability of the second event after the first event has occurred.

19. Example 3: Application A snack cart has 6 bags of pretzels and 10 bags of chips. Grant selects a bag at random, and then Iris selects a bag at random. What is the probability that Grant will select a bag of pretzels and Iris will select a bag of chips?

20. 1 • Grant chooses a bag of pretzels from 6 bags of pretzels and 10 bags of chips. • Iris chooses a bag of chips from 5 bags of pretzels and 10 bags of chips. Understand the Problem Example 3 Continued The answer will be the probability that a bag of chips will be chosen after a bag of pretzels is chosen. List the important information:

21. Make a Plan Iris chooses from: Grant chooses from: pretzels chips 2 Example 3 Continued Draw a diagram. After Grant selects a bag, the sample space changes. So the events are dependent. After Grant selects a bag, the sample space changes. So the events are dependent.

22. 3 Solve P(pretzel and chip) = P(pretzel) P(chip after pretzel) • The probability that Grant selects a bag of pretzels and Iris selects a bag of chips is . Example 3 Continued Grant selects one of 6 bags of pretzels from 16 total bags. Then Iris selects one of 10 bags of chips from 15 total bags.

23. Look Back 4 Example 3 Continued Drawing a diagram helps you see how the sample space changes. This means the events are dependent, so you can use the formula for probability of dependent events.

24. Check It Out! Example 3 A bag has 10 red marbles, 12 white marbles, and 8 blue marbles. Two marbles are randomly drawn from the bag. What is the probability of drawing a blue marble and then a red marble?

25. 1 Understand the Problem Check It Out! Example 3 Continued The answer will be the probability that a red marble will be chosen after a blue marble is chosen. List the important information: • A blue marble is chosen from a bag containing 10 red, 12 white, and 8 blue marbles. • Then a red marble is chosen from a bag of 10 red, 12 white, and 7 blue marbles.

26. Make a Plan First choice from: Second choice from: 2 Check It Out! Example 3 Continued Draw a diagram. After the first selection, the sample space changes. So the events are dependent.

27. 3 Solve P(blue and red) = P(blue) P(red after blue) • The probability that first a blue marble is selected and then a red marble is selected is . Check It Out! Example 3 Continued One of 8 blue marbles is selected from a total of 30 marbles. Then one of 10 red marbles is selected from the 29 remaining marbles.

28. 4 Check It Out! Example 3 Continued Look Back Drawing a diagram helps you see how the sample space changes. This means the events are dependent, so you can use the formula for probability of dependent events.

29. Lesson Quiz: Part I Tell whether each set of events is independent or dependent. Explain your answer. 1. flipping two different coins and each coin landing showing heads Independent; the flip of the first coin does not affect the sample space for the flip of the second coin. 2. drawing a red card from a standard deck of cards and not replacing it; then drawing a black card from the same deck of cards Dependent; there are fewer cards to choose from when drawing the black card.

30. Lesson Quiz: Part II 3. Eight cards are numbered from 1 to 8 and placed in a box. One card is selected at random and not replaced. Another card is randomly selected. What is the probability that both cards are greater than 5? 4. An experiment consists of randomly selecting a marble from a bag, replacing it, and then selecting another marble. The bag contains 3 yellow marbles and 2 white marbles. What is the probability of selecting a white marble and then a yellow marble?

31. Lesson Quiz: Part III 5. A number cube is rolled two times. What is the probability of rolling an even number first and then a number less than 3?